The **key difference between Lorentz gauge and Coulomb gauge **is that Lorentz gauge is related to the Minkowski space, whereas Coulomb gauge is related to the Euclidean space.

Generally, Minkowski space is a 4D (four-dimensional) real vector space. This is equipped with a nondegenerate, symmetric bilinear form. It also occurs on the tangent space at each point in spacetime. Euclidean space, on the other hand, is a fundamental in classical geometry. It is a 3 D (three-dimensional) space.

### CONTENTS

1. Overview and Key Difference

2. What is Lorentz Gauge

3. What is Coulomb Gauge

4. Lorentz Gauge vs Coulomb Gauge in Tabular Form

5. Summary – Lorentz Gauge vs Coulomb Gauge

## What is Lorentz Gauge?

Lorentz gauge is a partial gauge fixing of the electromagnetic vector potential. This concept was first described by Ludwig Lorenz. This term mainly has its applications in electromagnetism. Generally, we can use the Lorentz gauge in electromagnetism for the calculation of time-dependent electromagnetic fields through the related potentials.

Originally, when the work of Ludwig Lorenz was published, Maxwell did not receive it well. Thereafter, he eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation. This is because he was working in Coulomb gauge. More importantly, the Lorentz gauge is related to the Minkowski space.

## What is Coulomb Gauge?

Coulomb gauge is a type of gauge that is expressed in terms of instantaneous values of the fields and densities. It is also known as **a transverse gauge**. This concept is very useful in quantum chemistry and condensed matter physics. We can define it using gauge condition, or more precisely, using gauge fixing condition.

This Coulomb gauge is particularly useful in semi-classical calculations that come in quantum mechanics. Here, the vector potential is quantized, but Coulomb interaction is not. In the Coulomb gauge, we can express the potentials in terms of instantaneous values of the fields and densities.

Moreover, gauge transformations can retain the Coulomb gauge condition, which may be formed with gauge functions that satisfy the concept. However, in regions that are far from the electric charge of the scalar potential, the Coulomb gauge becomes zero, and we call it the radiation gauge. This electromagnetic radiation was first quantized in this gauge.

Furthermore, the Coulomb gauge admits a natural Hamiltonian formulation of the evolution equations (concerning the electromagnetic field) of the electromagnetic field interacting with a conserved current. This is an advantage of the quantization of the theory. More importantly, Coulomb gauge is related to the Euclidean space.

## What is the Difference Between Lorentz Gauge and Coulomb Gauge?

Lorentz gauge and Coulomb gauge are two concepts that are important in quantum chemistry. Lorentz gauge is a partial gauge fixing of the electromagnetic vector potential while Coulomb gauge is a type of gauge that is expressed in terms of instantaneous values of the fields and densities. The key difference between Lorentz gauge and Coulomb gauge is that the Lorentz gauge is related to the Minkowski space, whereas Coulomb gauge is related to the Euclidean space. Minkowski space is a 4D (four-dimensional) real vector space, while Euclidean space is 3 D (three-dimensional) space, which is also a fundamental of classical geometry.

Below is a summary of the difference between Lorentz gauge and Coulomb gauge in tabular form for side by side comparison.

## Summary – Lorentz Gauge vs Coulomb Gauge

We can distinguish the Lorentz gauge and Mikowskin gauge depending on the dimensions. The key difference between Lorentz gauge and Coulomb gauge is that Lorentz gauge is related to the Minkowski space, whereas Coulomb gauge is related to the Euclidean space. Minkowski space is a 4D (four-dimensional) real vector space, while Euclidean space is a fundamental of the classical geometry and is a 3 D (three-dimensional) space.

##### Reference:

1. Yann Picand, Dominique Dutoit. “Lorenz Gauge Condition.” *Sensagent*.

##### Image Courtesy:

1. “Coord system CA 0” By Jorge Stolfi – Own work (Public Domain) via Commons Wikimedia

2. “World line” By SVG version: K. Aainsqatsi at en.wikipediaOriginal PNG version: Stib at en.wikipedia – Transferred from en.wikipedia to Commons.(Original text: self-made) (CC BY-SA 3.0) via Commons Wikimedia

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